Application of Integrals

Let f:R be a twice differentiable function such that f(x+y)=f(x)f(y) for all x,y . If f'(0)=4a and f satisfies f''(x)-3af'(x)-f(x)=0, a>0, then the area of the region R=\(x,y) 0 y f(ax), 0 x 2\ is :

1. e2-1
2. e4+1
3. e2+1
4. e4-1

Answer

(1) e2-1

Solution

f(x+y)=f(x)f(y) with f'(0)=4a gives f(x)=e4ax.

f''-3af'-f=0 16a2-12a2-1=0 a= 12, so f(ax)=ex

The region is under y=ex for 0 x 2 (shaded):

[figure]

Area = 02ex dx=e2-1

Answer: (1)